Problem: All of the 3rd grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$4.50$ each for students, and the group paid $$55.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$14.00$ each for teachers and $$8.50$ each for students, and the group paid $$107.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+4.5y = 55}$ ${14x+8.5y = 107}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-9y = -110}$ ${14x+8.5y = 107}$ Add the top and bottom equations together. $ -0.5y = -3 $ $ y = \dfrac{-3}{-0.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {7x+4.5y = 55}$ to find $x$ ${7x + 4.5}{(6)}{= 55}$ $7x+27 = 55$ $7x = 28$ $x = \dfrac{28}{7}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {14x+8.5y = 107}$ and get the same answer for $x$ ${14x + 8.5}{(6)}{= 107}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.